Well, as far as I can tell the 6200 should be enough for SFM, it's supposed to support DirectX 9 shader model 3. If some apps tell you it only supports DX8 you might wannna update your graphics card drivers.

The "The game window is smaller than the sfm render resolution." error is caused by your screen resolution being too low for SFM's native 1280 X 720 render resolution. What's the make and model of your monitor? Perhaps you could push the resolution a bit higher.

Hm, you know I'm having doubts about the graphics card. Some sources seem to state the GeForce 6200 should be enough, but looking at the release dates, it seems it came out in 2004-2005, while the oldest GeForce 200 series card (minimum reqirement for SFM) came out in 2009... Wish I knew GeForce GPUs better...

EDIT:

Actually, I just remembered my parents' old computer had a GeForce 7300 in it, and I'm positive that hunk o' junk wouldn't run SFM in a million years. I'd wait for a second opinion since my knowledge on the subject is a tad limited, but I believe you have to buy a new computer if you ever hope to use SFM...

GeForce 6200 in a low-profile AGP form factorWith just 4 pixel pipelines, the 6200 series forms Nvidia's value/budget (low-end or entry level) product. The 6200 omits memory compression and SLI support, but otherwise offers similar rendering features as the 6600s. The later 6200 boards were based on the NV44 core (s), which is the final production silicon for the 6200 series.

the "The game window is smaller than the sfm render resolution." error thing is from having the 1080p set as your launch option. I get that everytime I launch SFM with 1080p enabled. but if i launch SFM without it I don't get the error. you get that because by default SFM (and specifically game mode (F11)) is set to 720p.